Is $W_{0}^{1,p}\left(\Omega\right)$ compactly embedded in $L^{\infty}\left(\Omega\right)$?

Question

Let $\Omega$ be an open bounded smooth domain in $\mathbb{R}^{N}$. Let $p>N$, is $W_{0}^{1,p}\left(\Omega\right)$ compactly embedded in $L^{\infty}\left(\Omega\right)$? In some textbook such as Sobolev Spaces (Adams-1975), it was only said that if $p>N$ , $W_{0}^{1,p}\left(\Omega\right)$ compactly embedded in $C\left(\overline{\Omega}\right)$

Answer 1

Yes, and here's why:

1. A $W^{1,p}_0$ functions extends (by $0$), to a $W^{1,p}$ function on a larger ball.
2. The Morrey oscillation inequality gives Hölder continuity with a bound depending only on the $W^{1,p}$ norm.
3. By the Arzelà-Ascoli theorem, the unit ball of $W^{1,p}_0$ is precompact in the $L^\infty$ norm.